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 A121574 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)). 2

%I

%S 1,2,1,4,5,1,8,16,8,1,16,44,37,11,1,32,112,134,67,14,1,64,272,424,305,

%T 106,17,1,128,640,1232,1168,584,154,20,1,256,1472,3376,3992,2641,998,

%U 211,23,1,512,3328,8864,12592,10442,5221,1574,277,26,1

%N Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

%C Row sums are A006190(n+1); diagonal sums are A077939.

%C Inverse is A121575.

%C A generalized Delannoy number triangle.

%C Antidiagonal sums are A002478. - _Philippe Deléham_, Nov 10 2011.

%H G. C. Greubel, <a href="/A121574/b121574.txt">Rows n = 0..100 of triangle, flattened</a>

%F Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).

%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - _Philippe Deléham_, Nov 10 2011

%e Triangle begins

%e 1;

%e 2, 1;

%e 4, 5, 1;

%e 8, 16, 8, 1;

%e 16, 44, 37, 11, 1;

%e 32, 112, 134, 67, 14, 1;

%e 64, 272, 424, 305, 106, 17, 1;

%p T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # _Muniru A Asiru_, Nov 02 2018

%t Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 02 2018 *)

%o (PARI) for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ _G. C. Greubel_, Nov 02 2018

%o (MAGMA) [[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 02 2018

%o (GAP) T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # _Muniru A Asiru_, Nov 02 2018

%Y Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Aug 08 2006

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Last modified April 11 08:10 EDT 2021. Contains 342886 sequences. (Running on oeis4.)